First, in your pan (10" works well) melt some butter and olive oil together. Or use some bacon drippings. A couple of tablespoons should do it. Your filet has no fat on it so you need to add some to the pan. You should salt and pepper both sides. Then brown them nicely on medium high heat. You can tell the browning is complete when the meat doesn't stick to the pan. Like I said, cast iron is best.
Pre-heat your oven to 400 degrees. When your filets are brown, pop them in the oven. Be careful and use thick oven mitts for this. Your pan till be very HOT. Set your timer for 5 minutes. At this point you'll begin to check the temperature. If you like your meat on the rare side you'll pull them at 145 degrees on your thermometer. Decide which filet is yours and cut into it a bit. If it's too rare, put them back in the oven for another 3 minutes. Check your temperature each time. Since pork is done at around 160, you'll probably hit your sweet spot between 145 and 155. Otherwise, the meat is gray and the juices are dried up. Remember, you check a steak by cutting into it and serving it with that side down. No one will know!
My dinner last night. A perfectly done. Filet mignon, brussel sprouts with bacon, rice pilaf. Perfect!
Stick with me and we'll roast a chicken to perfection outside on the gas grill.